In the field of computational structural acoustics, the problem of efficiently modeling the acoustic field in large exterior domains has remained a difficult challenge for over a quarter century. Many techniques have evolved. Those historically receiving greatest attention in the literature are: the boundary integral equation method or, as it is often called, the boundary element method (BEM), which is based on the surface Helmholtz integral equation applied to the surface of the structure; and the infinite element method (IEM), which is based on the Helmholtz differential equation (the reduced wave equation), applied to semi-infinite sectors of the domain (which are the infinite elements) that are exterior to an artificial boundary surrounding the structure, with finite elements between the structure and the boundary.
The BEM is the method of choice for most researchers and code developers. Its primary advantage is that it is based on a mathematically rigorous formulation, namely, the Helmholtz integral representation, that (i) satisfies the Sommerfield radiation condition and (ii) represents the exact solution throughout the exterior domain. In addition, its reduction of the dimensionality of the problem by one has long been thought to be a significant computational advantage. However, we have found that this is actually disadvantageous because of the much greater bandwidth of the equations, due to the inherent global connectivity of the method.
By contrast, infinite elements were hitherto never intended to represent the exact solution, or even an approximate solution, within the element itself. Based on ad hoc physical assumptions or asymptotic theories, they were only meant to provide an approximate non-reflecting condition on an artificial boundary surrounding the structure, thereby enabling a sufficiently accurate solution to be obtained in the finite region interior to the boundary. The solution exterior to the boundary is then usually obtained from the exterior Helmholtz integral, using the pressure and velocity data computed on the surface of the structure.
Two types of infinite elements have previously been used for acoustic applications: the exponential decay and the "mapped" element.
The exponential decay element approximates the spatial decay of the acoustic pressure amplitude by an exponential, pe.sup.-.gamma.r e.sup.-ikr , where .gamma. is an empirically adjusted positive number and r is any coordinate that extends to infinity. Because this decay function is inconsistent with acoustical theory, the accuracy rapidly deteriorates away from the artificial boundary.
The mapped element is based on the asymptotic form of the field for large r, using the leading terms of the lower order spherical Hankel functions, namely, ##EQU1## Because of the mapping, the element was intended to be usable in any shape and orientation, i.e., not necessarily aligned with spherical coordinate surfaces, thereby permitting the "radial" sides of different elements to emanate from different spherical origins.
However, we have found that for this representation to converge, (i) the elements must lie outside the smallest sphere circumscribing the structure, and (ii) the sides of all elements in a mesh must conform to radial lines emanating from the origin of the single r coordinate in the expansion. Since the fluid region between the circumscribing sphere and the structure must be meshed with finite-size acoustic elements, the total acoustic mesh would become very large and therefore computationally very inefficient for structures of large aspect ratio. Thus, the element is practical only for "chunky" structures, i.e., those that can be closely circumscribed by a sphere. This is a serious limitation for many applications.
An additional problem of the mapped element is that element generation is relatively expensive. Moreover, generation of the matrices for the mapped element requires inversion of an ill-conditioned matrix. This causes quadrature problems at very high frequencies.
In U.S. Pat. No. 5,604,891, entitled "A 3-D Acoustic Infinite Element Based On A Prolate Spheroidal Multipole Expansion," assigned to Applicants' assignee, we described a new infinite element based on a multipole expansion in prolate spheroidal coordinates. This infinite element is also described in D. S. Burnett et al., "A Three-Dimensional Acoustic Infinite Element Based on a Prolate Spheroidal Multipole Expansion," Journal of the Acoustical Society of America 96 (November 1994) 2798-2816.
We have found that the use of this infinite element leads to an extraordinary improvement in efficiency in the acoustic modeling of elongate structures (i.e., structures in which one dimension is much longer than the other two). For example, careful benchmark comparisons on structures with a 10:1 aspect ratio have revealed computational speeds that are over 400 times faster than the BEM, to achieve the same answer to the same accuracy. The more elongate the structure, the greater the increase in speed. Therefore, it is expected that speed-enhancement ratios will extend into the thousands for even longer structures.
Since prolate spheroids include spheres as a limiting case, the prolate spheroidal infinite element can also efficiently model structures that are "chunky," i.e., all three dimensions comparable. In such cases the computational speeds relative to the BEM are still dramatic, but not as extraordinary as for elongate structures.
If a structure has a flat, i.e., "disk-like," shape (one dimension much shorter than the other two), the prolate infinite element becomes less appropriate, through probably still faster than the BEM. For such structures, it is more appropriate to use an oblate spheroidal infinite elment.
In U.S. Pat. No. 5,604,893, entitled "A 3-D Acoustic Infinite Element Based on an Oblate Spheroidal Multipole Expansion," assigned to Applicants' assignee, we described a new infinite element based on a multipole expansion in oblate spheroidal coordinates. Like the prolate element, this oblate infinite element leads to extraordinary efficiencies in the acoustic modeling of certain structures, namely, generally flat, disk-like structures.
There remains a third class of structural shapes that is characterized by all three dimensions being markedly different; i.e., one dimension is much longer than the second, which in turn is much longer than the third. Such structures could be described as being both long and flat, i.e., strip-like or ribbon-like. Although either the prolate or oblate spheroidal infinite elements would provide extraordinary speed increases for this class, there is another coordinate system that would provide still greater (that is, significantly greater) speed increases: it is the ellipsoidal coordinate system, and it has the additional advantage of including prolate and oblate spheroidal coordinates as special cases. Thus, an infinite element based on the ellipsoidal coordinate system is the most general of all, providing extraordinary computational efficiency for structures of any shape, no matter how disparate their dimensions might be.